Custom Mesh Texture - Best Method?
You could do it with texture paint, you would have control on how your spots appear exactly, or, as John says, you could do it procedurally.

If you choose this second option, here is a try. Set this nodes chain: Texture Coordinate (Object output) > Mapping node > Texture node (here Musgrave) > ColorRamp > Mix Shader. Mix your 2 colors (carrot color and spot color) in the Mix Shader. Play with the Mapping scale values to squish the spots on the right axis and with the parameters of the other nodes to give them the shape you want. At the end of the chain mix with a Glossy, and maybe also with a Subsurface Scattering to give it a bit of transparency.